Found out the problem!
Had an older project with an AtMega4808 with a display and just to verify that the 'broken' displays do not work, tested one of them with it. It worked! After a couple of hours of comparing the codes (the current one is with Raspberry Pico), I found out that the SPI settings were not identical. While the datasheet of the ST7789 implies POL=0 PHA=0, the CS-less connection of these cheap displays seems to require POL=1 PHA=1. HOWEVER, half of the displays also work reliably with POL=1 PHA=0 (with Pico, at least). So, it was this that got me stuck - since some of the displays did work 100% reliably, it never occurred to me that the settings could be so wrong.
So,
1) My bad
2) There really are 2 versions around, since the behavior is 100% repeatable and consistent for each individual display
3) The ditching of CS is an idiotic place to save 1 pin worth of cost
4) A working system can easily be mistaken for a correct system